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Ergodic theoretic methods in group h...

Loh, Clara.

Ergodic theoretic methods in group homologya minicourse on L2-Betti numbers in group theory /

Record Type:	Electronic resources : Monograph/item
Title/Author:	Ergodic theoretic methods in group homologyby Clara Loh.
Reminder of title:	a minicourse on L2-Betti numbers in group theory /
Author:	Loh, Clara.
Published:	Cham :Springer International Publishing :2020.
Description:	ix, 114 p. :ill., digital ;24 cm.
Contained By:	Springer eBooks
Subject:	Group theory.
Online resource:	https://doi.org/10.1007/978-3-030-44220-0
ISBN:	9783030442200$q(electronic bk.)

Ergodic theoretic methods in group homologya minicourse on L2-Betti numbers in group theory /
Loh, Clara.

Ergodic theoretic methods in group homologya minicourse on L2-Betti numbers in group theory /[electronic resource] :by Clara Loh. - Cham :Springer International Publishing :2020. - ix, 114 p. :ill., digital ;24 cm. - SpringerBriefs in mathematics,2191-8198. - SpringerBriefs in mathematics..

0 Introduction -- 1 The von Neumann dimension -- 2 L2-Betti numbers -- 3 The residually finite view: Approximation -- 4 The dynamical view: Measured group theory -- 5 Invariant random subgroups -- 6 Simplicial volume -- A Quick reference -- Bibliography -- Symbols -- Index.

This book offers a concise introduction to ergodic methods in group homology, with a particular focus on the computation of L2-Betti numbers. Group homology integrates group actions into homological structure. Coefficients based on probability measure preserving actions combine ergodic theory and homology. An example of such an interaction is provided by L2-Betti numbers: these invariants can be understood in terms of group homology with coefficients related to the group von Neumann algebra, via approximation by finite index subgroups, or via dynamical systems. In this way, L2-Betti numbers lead to orbit/measure equivalence invariants and measured group theory helps to compute L2-Betti numbers. Similar methods apply also to compute the rank gradient/cost of groups as well as the simplicial volume of manifolds. This book introduces L2-Betti numbers of groups at an elementary level and then develops the ergodic point of view, emphasising the connection with approximation phenomena for homological gradient invariants of groups and spaces. The text is an extended version of the lecture notes for a minicourse at the MSRI summer graduate school "Random and arithmetic structures in topology" and thus accessible to the graduate or advanced undergraduate students. Many examples and exercises illustrate the material.

ISBN: 9783030442200$q(electronic bk.)

Standard No.: 10.1007/978-3-030-44220-0doiSubjects--Topical Terms:

189579
Group theory.

LC Class. No.: QA174.2 / .L865 2020

Dewey Class. No.: 512.2

Ergodic theoretic methods in group homologya minicourse on L2-Betti numbers in group theory /
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505 0 $a 0 Introduction -- 1 The von Neumann dimension -- 2 L2-Betti numbers -- 3 The residually finite view: Approximation -- 4 The dynamical view: Measured group theory -- 5 Invariant random subgroups -- 6 Simplicial volume -- A Quick reference -- Bibliography -- Symbols -- Index.
520 $a This book offers a concise introduction to ergodic methods in group homology, with a particular focus on the computation of L2-Betti numbers. Group homology integrates group actions into homological structure. Coefficients based on probability measure preserving actions combine ergodic theory and homology. An example of such an interaction is provided by L2-Betti numbers: these invariants can be understood in terms of group homology with coefficients related to the group von Neumann algebra, via approximation by finite index subgroups, or via dynamical systems. In this way, L2-Betti numbers lead to orbit/measure equivalence invariants and measured group theory helps to compute L2-Betti numbers. Similar methods apply also to compute the rank gradient/cost of groups as well as the simplicial volume of manifolds. This book introduces L2-Betti numbers of groups at an elementary level and then develops the ergodic point of view, emphasising the connection with approximation phenomena for homological gradient invariants of groups and spaces. The text is an extended version of the lecture notes for a minicourse at the MSRI summer graduate school "Random and arithmetic structures in topology" and thus accessible to the graduate or advanced undergraduate students. Many examples and exercises illustrate the material.
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